一类状态脉冲反馈控制下的营养盐–浮游植物动力系统的相关动力学分析Related Dynamic Analysis of a Nutrient-Phytoplankton Dynamic System under State Impulsive Feedback Control

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On the basis of state impulsive feedback control theory and nonlinear dynamic system theory, the Michaelis-Menten functional response function is introduced to describe the interaction mechanism of phytoplankton and nutrients in the process of dynamic modeling; a nutrient-phytoplankton dynamic system under state impulsive feedback control has been structured. Some qualitative analysis of the dynamic system has been investigated to establish some theoretical criterions for the existence, uniqueness and asymptotic stability of the order-1 periodic solution. The research work can provide a theoretical support for the comprehensive study of the application of state impulsive feedback control theory to the prevention and control of eutrophication in water body.

1. 引言

$\left\{\begin{array}{l}\begin{array}{l}\frac{\text{d}x}{\text{d}t}=I-\frac{\alpha xy}{\beta +x}-qx\\ \frac{\text{d}y}{\text{d}t}=\frac{\epsilon \alpha xy}{\beta +x}-my\end{array}\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y (1)

2. 模型和准备

$\phi :\left(x,y\right)\in M\to \left(x,\left(1-p\right)y\right)\in {R}_{+}^{2}$ ,

$N=\phi \left(M\right)=\left\{\left(x,y\right)\in {R}_{+}{}^{2}|y=\left(1-p\right)h,x\ge 0\right\}$ .

$d\left(z\left(t\right),O\left({z}_{0},{t}_{0}\right)\right)<\epsilon$

3. 正周期解的存在性与稳定性

${C}_{3}$ 出发的轨迹交脉冲集 $M$ 于点 ${C}_{4}$ ，接着跳到后继点 ${C}_{5}$ ，而 ${C}_{5}$ 必定位于 ${C}_{3}$ 的右边，因此有 ${x}_{{C}_{5}}>{x}_{{C}_{3}}$ 。因此可得两个序列：

${x}_{{C}^{\prime }},{x}_{{C}_{2}},{x}_{{C}_{4}},\cdots ,{x}_{{C}_{2n}},\cdots \in M,$

${x}_{{C}_{1}},{x}_{{C}_{3}},{x}_{{C}_{5}},\cdots ,{x}_{{C}_{2n-1}},\cdots \in N.$

${x}_{B\text{'}},{x}_{{B}_{2}},{x}_{{B}_{4}},\cdots ,{x}_{{B}_{2n}},\cdots \in M,$

${x}_{{B}_{1}},{x}_{{B}_{3}},{x}_{{B}_{5}},\cdots ,{x}_{{B}_{2n-1}},\cdots \in N.$

$\left({x}_{C},{x}_{B}\right)\supset \left({x}_{{C}_{1}},{x}_{{B}_{1}}\right),\text{\hspace{0.17em}}d\left(C,B\right)\supset d\left({C}_{1},{B}_{1}\right).$

${x}_{{C}_{1}}<{x}_{{C}_{3}}<\cdot \cdot \cdot <{x}_{{C}_{2n-1}}<\cdot \cdot \cdot <{x}_{H}<{x}_{{B}_{2n-1}}<\cdot \cdot \cdot <{x}_{{B}_{3}}<{x}_{{B}_{1}},$

$\underset{n\to \infty }{\mathrm{lim}}\text{\hspace{0.17em}}d\left({C}_{2n-1},{B}_{2n-1}\right)=0$

$\underset{n\to \infty }{\mathrm{lim}}\text{\hspace{0.17em}}{x}_{{C}_{2n-1}}={x}_{H},\text{\hspace{0.17em}}\underset{n\to \infty }{\mathrm{lim}}\text{\hspace{0.17em}}{x}_{{B}_{2n-1}}={x}_{H}.$

$H$ 足够小的邻域内，选取任意一点 ${Q}_{0}$ (不同于 $H$ )，不失一般性，假设 ${x}_{C}<{x}_{{Q}_{0}}<{x}_{H}$ (否则 ${x}_{H}<{x}_{{Q}_{0}}<{x}_{B}$ ，证明方法同理)。

1) ${Q}_{k}$ 位于 ${C}_{2l+2k+1}$${C}_{2l+2k+3}$ 之间；

2) ${x}_{{C}_{2l+2k+1}}<{x}_{{Q}_{k}}<{x}_{{C}_{2l+2k+3}}$

$0<{x}_{j+1}<{x}_{i+1}<{x}^{*}$ (2)

(a) 若 ${x}_{0}<{x}_{1}$ ，由(2)得 ${x}_{1}>{x}_{2}$ ，在这种情形下， ${x}_{0},{x}_{1},{x}_{2}$ 的关系有如下两种情况：

(i) ${x}_{2}<{x}_{0}<{x}_{1}$

${x}_{2}<{x}_{0}<{x}_{1}$ ，则有 ${x}_{3}>{x}_{1}>{x}_{2}$ 。由(2)重复以上的过程得到：

$0<\cdots <{x}_{2n}<\cdots <{x}_{2}<{x}_{0}<{x}_{1}<\cdots <{x}_{2n+1}<\cdots <{x}^{*}$ .

(ii) ${x}_{0}<{x}_{2}<{x}_{1}$

${x}_{0}<{x}_{2}<{x}_{1}$ ，类比(i)，得到：

${x}_{0}<{x}_{2}<\cdots <{x}_{2n}<\cdots <{x}_{2n+1}<\cdots <{x}_{3}<{x}_{1}<{x}^{*}$ .

(b) 若 ${x}_{0}>{x}_{1}$ ，由(2)可得 ${x}_{1}<{x}_{2}$ ，这种情形下， ${x}_{0},{x}_{1},{x}_{2}$ 的关系有如下两种：

(i) ${x}_{1}<{x}_{0}<{x}_{2}$

${x}_{1}<{x}_{0}<{x}_{2}$ ，则有 ${x}_{2}>{x}_{1}>{x}_{3}$ ，由(2)，并重复以上过程得到：

$0<\cdots <{x}_{2n+1}<\cdots <{x}_{1}<{x}_{0}<{x}_{2}<\cdots <{x}_{2n}<\cdots <{x}^{*}$ .

(ii) ${x}_{1}<{x}_{2}<{x}_{0}$

${x}_{1}<{x}_{2}<{x}_{0}$ ，类比(i)得到：

$0<{x}_{1}<\cdots <{x}_{2n+1}<\cdots <{x}_{2n}<\cdots <{x}_{2}<{x}_{0}<{x}^{*}$ .

4. 结论

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